What is the difference between learning without forgetting and transfer learning?
Learning without Forgetting (LwF) is an incremental learning (sometimes also called continual or lifelong learning) technique for neural networks, which is a machine learning technique that attempts to avoid catastrophic forgetting. There are several incremental learning approaches. LwF is an incremental learning approach based on the concept of regularization. In section 3.2 of the paper Continual lifelong learning with neural networks: A review (2019), by Parisi et al., other regularisation-based continual learning techniques are described. 

LwF could be seen as a combination of distillation networks and fine-tuning, which refers to the re-training with a low learning rate (which is a very rudimentary technique to avoid catastrophically forgetting the previously learned knowledge) an already trained model $\mathcal{M}$ with new and (usually) more specific dataset, $\mathcal{D}_{\text{new}}$, with respect to the dataset, $\mathcal{D}_{\text{old}}$, with which you originally trained the given model $\mathcal{M}$. 

LwF, as opposed to other continual learning techniques, only uses the new data, so it assumes that past data (used to pre-train the network) is unavailable. The paper Learning without Forgetting goes into the details of the technique and it also describes the concepts of feature extraction, fine tuning and multitask learning, which are related to incremental learning techniques.

What is the difference between LwF and transfer learning? LwF is a combination of distillation networks and fine-tuning, which is a transfer learning technique, which is a special case of incremental learning, where the old and new tasks are different, while, in general, in incremental learning, the old and new tasks can also be the same (which is called domain adaptation).